BIRS Workshop Lecture Videos

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BIRS Workshop Lecture Videos

Patching and self-duality Manning, Jeffrey


I will describe a method for determining the structure of a "patched module" arising from the Taylor-Wiles-Kisin method in the case when the local Galois deformation rings are not formally smooth. It was observed by Diamond that commutative algebra techniques (specifically the Auslanderâ Buchsbaum formula) can be used to show that a patched module is free, implying a mod $\ell$ multiplicity one statement, in the specific case when the relevant local deformation rings are all formally smooth. I will present a new method for determining the structure of certain patched modules which works when the local deformation rings are not formally smooth, provided these rings can still be computed explicitly, by exploiting the natural self-duality of many common patched modules. This method can be used to explicitly compute patched modules even in cases when they are not free. Using this (and Shotton's computations of local deformation rings in the $\ell \neq p$ case), I will compute the patched module arising from the cohomology of a Shimura curve, to prove a mod $\ell$ "multiplicity $2^k$" statement in the minimal level case, generalizing a result of Ribet. The precise computation of this patched module also yields additional information about the Hecke module structure of the cohomology of a Shimura curve, which among other things has applications to the study of congruence modules. Time permitting, I will also describe how this method can be extended to the "$l_0>0$" patching situation introduced by Calegari and Geraghty, and describe partial work towards generalizing it to the case of n dimensional representations associated to the cohomology of unitary Shimura varieties.

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